Subgroups — #14315137

Order 1920.

324 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91227	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#91219	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65014	C₂	1	1	yes	maximal, characteristic
4	#11746782	C₂	1	1	yes	maximal, characteristic, complemented
5	#11746759	C₂	1	1	yes	maximal, characteristic, complemented
6	#11746758	C₂	1	1	yes	maximal, characteristic, complemented
7	#11746756	C₂	1	1	yes	maximal, characteristic, complemented
8	#10603906	—	3	1	no	maximal, Hall
9	C₃×C₄×C₂²⋊D₅	C₄	1	1	yes	characteristic
10	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
12	#91227	—	2	2	no	—
13	C₂×C₃⋊#674	C₂²	1	1	yes	characteristic
14	C₂×C₃⋊#674	C₂²	1	1	yes	characteristic
15	C₂×C₃⋊#674	C₂²	1	1	yes	characteristic
16	C₂×C₃⋊#674	C₂²	1	1	yes	characteristic
17	#91219	—	2	2	no	—
18	C₃×C₅×C₂²⋊C₈	C₄	1	1	yes	characteristic
19	C₃×#3952	C₄	1	1	yes	characteristic
20	C₃×#3946	C₄	1	1	yes	characteristic
21	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
22	#70858	—	5	1	no	maximal, Hall
23	C₂×#3998	—	3	1	no	—
24	C₂×#3990	—	3	1	no	—
25	#65014	D₃	1	1	yes	characteristic
26	C₂²⋊#674	—	3	1	no	—
27	#64888	—	3	1	no	—
28	#64887	—	3	1	no	—
29	#64885	—	3	1	no	—
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₂²⋊D₅	C₈	1	1	yes	characteristic
32	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
35	C₃×C₂²⋊Dic₅	C₈	1	1	yes	characteristic
36	C₃×#685	C₈	1	1	yes	characteristic
37	C₃×#683	C₈	1	1	yes	characteristic
38	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×#168	—	2	1	no	—
40	C₂×C₃×#168	—	2	1	no	—
41	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃×#168	C₂³	1	1	yes	characteristic
43	C₃×C₈×D₅	—	2	1	no	—
44	C₃×C₈×D₅	—	2	1	no	—
45	C₃⋊#674	—	2	1	no	—
46	C₃⋊#674	—	2	1	no	—
47	C₃⋊#674	—	2	1	no	—
48	C₃⋊#674	D₄	1	2	yes	complemented
49	C₃×C₈×D₄	—	5	1	no	—
50	C₂×#812	—	5	1	no	—
51	C₂×#812	—	5	1	no	—
52	#4549	—	5	1	no	—
53	#4549	—	5	1	no	—
54	C₄⋊(C₃⋊C₁₆)	—	5	1	no	—
55	C₄×C₃⋊C₁₆	—	5	1	no	—
56	C₄×C₂²⋊D₅	Dic₃	1	1	yes	characteristic
57	C₂²×#168	C₂×D₃	1	1	yes	characteristic
58	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
59	#3998	—	6	2	no	—
60	C₂×#674	—	3	1	no	—
61	C₂×#674	—	3	1	no	—
62	C₂×#674	—	3	1	no	—
63	C₂×#674	—	3	1	no	—
64	#3990	—	6	2	no	—
65	C₅×C₂²⋊C₈	Dic₃	1	1	yes	characteristic
66	#3952	Dic₃	1	1	yes	characteristic
67	#3946	Dic₃	1	1	yes	characteristic
68	C₈×Dic₅	C₂×D₃	1	1	yes	characteristic
69	#2169	—	15	1	no	Sylow
70	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
71	C₂²×C₃×D₅	C₂×C₈	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	—	2	1	no	—
73	C₂×C₃×C₄×C₅	—	2	1	no	—
74	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₃×C₂²⋊D₅	—	2	2	no	—
77	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
78	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
79	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
80	C₃×C₄×D₅	—	2	1	no	—
81	C₃×C₄×D₅	—	2	1	no	—
82	C₃×C₅×C₈	—	2	1	no	—
83	C₃×C₅×C₈	—	2	1	no	—
84	C₃×#168	—	2	1	no	—
85	C₃×#168	C₂×D₄	1	1	yes	characteristic
86	C₃×#168	#41	1	1	yes	characteristic
87	C₂²×C₃×C₈	—	5	1	no	—
88	C₂²×C₃×C₈	—	5	1	no	—
89	C₃×C₄×D₄	—	5	1	no	—
90	C₃×C₄⋊C₈	—	5	1	no	—
91	C₃×C₂²⋊C₈	—	5	1	no	—
92	C₃×C₂²⋊C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
93	C₃×C₄×C₈	—	5	1	no	—
94	#812	—	10	2	no	—
95	#812	—	10	2	no	—
96	C₂×C₃⋊C₁₆	—	5	1	no	—
97	C₂×C₃⋊C₁₆	—	5	1	no	—
98	C₂×C₃⋊C₁₆	—	5	1	no	—
99	C₂×C₃⋊C₁₆	—	5	1	no	—
100	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
101	C₂×C₂²⋊D₅	C₃⋊C₈	1	1	yes	characteristic
102	C₂×C₄×D₅	C₂×Dic₃	1	1	yes	characteristic
103	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
104	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
105	C₂²⋊Dic₅	C₃⋊C₈	1	1	yes	characteristic
106	#685	C₃⋊C₈	1	1	yes	characteristic
107	#683	C₃⋊C₈	1	1	yes	characteristic
108	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
109	C₂×#168	—	2	1	no	—
110	C₂×#168	—	2	1	no	—
111	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
112	C₂×#168	C₂²×D₃	1	1	yes	characteristic
113	C₈×D₅	—	2	1	no	—
114	C₈×D₅	—	2	1	no	—
115	#674	—	3	2	no	—
116	#674	—	6	1	no	—
117	#674	—	6	1	no	—
118	#674	—	6	1	no	—
119	C₂×M₃₂	—	15	1	no	—
120	C₂×M₃₂	—	15	1	no	—
121	C₈×D₄	—	5	1	no	—
122	C₄⋊C₁₆	—	15	1	no	—
123	C₂²⋊C₁₆	—	15	1	no	—
124	C₂²⋊C₁₆	—	15	1	no	—
125	C₄×C₁₆	—	15	1	no	—
126	C₂²×C₃×C₅	—	2	1	no	—
127	C₂²×C₃×C₅	—	2	1	no	—
128	C₂²×C₃×C₅	—	2	1	no	—
129	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₃×D₅	—	2	1	no	—
131	C₂×C₃×D₅	—	2	1	no	—
132	C₂×C₃×D₅	—	2	1	no	—
133	C₃×C₄×C₅	—	2	1	no	—
134	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
135	C₃×C₄×C₅	#131	1	1	yes	characteristic
136	C₃×Dic₅	—	2	1	no	—
137	C₃×Dic₅	—	2	1	no	—
138	C₃×Dic₅	M₃₂	1	2	yes	complemented
139	C₂×C₃×D₄	—	5	1	no	—
140	C₂²×C₃×C₄	—	5	1	no	—
141	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
142	C₂×C₃×C₈	—	5	1	no	—
143	C₂×C₃×C₈	—	5	1	no	—
144	C₂×C₃×C₈	—	10	1	no	—
145	C₂×C₃×C₈	—	10	1	no	—
146	C₂×C₃×C₈	—	10	1	no	—
147	C₂×C₃×C₈	—	10	1	no	—
148	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
149	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
150	C₃×C₄⋊C₄	—	5	1	no	—
151	C₃×C₂²⋊C₄	—	5	1	no	—
152	C₃×C₂²⋊C₄	—	5	1	no	—
153	C₃×C₄²	—	5	1	no	—
154	C₃⋊C₁₆	—	5	2	no	—
155	C₃⋊C₁₆	—	10	1	no	—
156	C₃⋊C₁₆	—	10	1	no	—
157	C₃⋊C₁₆	—	10	1	no	—
158	C₂³×C₅	C₂×C₃⋊C₈	1	1	yes	characteristic
159	C₂²×D₅	C₂×C₃⋊C₈	1	1	yes	characteristic
160	C₂×C₄×C₅	—	2	1	no	—
161	C₂×C₄×C₅	—	2	1	no	—
162	C₂×C₄×C₅	C₂×C₃⋊C₈	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
164	C₂²⋊D₅	—	2	2	no	—
165	C₂×Dic₅	C₂×C₃⋊C₈	1	1	yes	characteristic
166	C₂×Dic₅	C₂×C₃⋊C₈	1	1	yes	characteristic
167	C₂×Dic₅	C₂×C₃⋊C₈	1	1	yes	characteristic
168	C₄×D₅	—	2	1	no	—
169	C₄×D₅	—	2	1	no	—
170	C₅×C₈	—	2	1	no	—
171	C₅×C₈	—	2	1	no	—
172	#168	—	2	1	no	—
173	#168	D₃×D₄	1	1	yes	characteristic
174	#168	#237	1	1	yes	characteristic
175	C₂²×C₈	—	5	1	no	—
176	C₂²×C₈	—	5	1	no	—
177	C₄×D₄	—	5	1	no	—
178	M₃₂	—	30	2	no	—
179	M₃₂	—	30	2	no	—
180	C₂×C₁₆	—	15	1	no	—
181	C₂×C₁₆	—	15	1	no	—
182	C₂×C₁₆	—	15	1	no	—
183	C₂×C₁₆	—	15	1	no	—
184	C₄⋊C₈	—	5	1	no	—
185	C₂²⋊C₈	—	5	1	no	—
186	C₂²⋊C₈	ΓL(1,𝔽₁₆)	1	1	yes	characteristic
187	C₄×C₈	—	5	1	no	—
188	C₂×C₃×C₅	—	4	1	no	—
189	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
190	C₂×C₃×C₅	C₂×M₃₂	1	1	yes	characteristic
191	C₂×C₃×C₅	#504	1	1	yes	characteristic
192	C₃×D₅	—	4	1	no	—
193	C₂³×C₃	—	5	1	no	—
194	C₂³×C₃	D₅⋊C₈	1	1	yes	characteristic
195	C₃×D₄	—	10	2	no	—
196	C₂×C₃×C₄	—	2	1	no	—
197	C₂×C₃×C₄	—	2	1	no	—
198	C₂×C₃×C₄	—	5	1	no	—
199	C₂×C₃×C₄	—	5	1	no	—
200	C₂×C₃×C₄	—	5	1	no	—
201	C₂×C₃×C₄	—	10	1	no	—
202	C₂×C₃×C₄	—	10	1	no	—
203	C₂×C₃×C₄	D₅⋊C₈	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
205	C₃×C₈	—	2	1	no	—
206	C₃×C₈	—	2	1	no	—
207	C₃×C₈	—	5	1	no	—
208	C₃×C₈	—	5	1	no	—
209	C₃×C₈	—	10	1	no	—
210	C₂²×C₅	—	2	1	no	—
211	C₂²×C₅	—	2	1	no	—
212	C₂²×C₅	—	2	1	no	—
213	C₂²×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
214	C₂×D₅	—	2	1	no	—
215	C₂×D₅	—	2	1	no	—
216	C₂×D₅	—	2	1	no	—
217	C₄×C₅	—	2	1	no	—
218	C₄×C₅	D₄×Dic₃	1	1	yes	characteristic
219	C₄×C₅	#948	1	1	yes	characteristic
220	Dic₅	—	2	1	no	—
221	Dic₅	—	2	1	no	—
222	Dic₅	#812	1	2	yes	complemented
223	C₂×D₄	—	5	1	no	—
224	C₂²×C₄	—	5	1	no	—
225	C₂²×C₄	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
226	C₂×C₈	—	5	1	no	—
227	C₂×C₈	—	5	1	no	—
228	C₂×C₈	—	10	1	no	—
229	C₂×C₈	—	10	1	no	—
230	C₂×C₈	—	10	1	no	—
231	C₂×C₈	—	10	1	no	—
232	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
233	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
234	C₄⋊C₄	—	5	1	no	—
235	C₂²⋊C₄	—	5	1	no	—
236	C₂²⋊C₄	—	5	1	no	—
237	C₄²	—	5	1	no	—
238	C₁₆	—	15	2	no	—
239	C₁₆	—	30	1	no	—
240	C₁₆	—	30	1	no	—
241	C₁₆	—	30	1	no	—
242	C₃×C₅	#2169	1	1	yes	characteristic, complemented, Hall
243	C₂²×C₃	—	2	1	no	—
244	C₂²×C₃	—	2	1	no	—
245	C₂²×C₃	—	2	1	no	—
246	C₂²×C₃	—	10	1	no	—
247	C₂²×C₃	—	10	1	no	—
248	C₂²×C₃	—	10	1	no	—
249	C₂²×C₃	C₂×D₅⋊C₈	1	1	yes	characteristic
250	C₃×C₄	—	2	1	no	—
251	C₃×C₄	—	5	2	no	—
252	C₃×C₄	—	10	1	no	—
253	C₃×C₄	—	10	1	no	—
254	C₃×C₄	#4131	1	1	yes	characteristic
255	C₃×C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
256	C₂×C₅	—	4	1	no	—
257	C₂×C₅	D₄×C₃⋊C₈	1	1	yes	characteristic
258	C₂×C₅	C₂×#812	1	1	yes	characteristic
259	C₂×C₅	#5140	1	1	yes	characteristic
260	D₅	—	4	1	no	—
261	C₂³	—	5	1	no	—
262	C₂³	D₅⋊(C₃⋊C₈)	1	1	yes	characteristic
263	D₄	—	10	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	5	1	no	—
267	C₂×C₄	—	5	1	no	—
268	C₂×C₄	—	5	1	no	—
269	C₂×C₄	—	10	1	no	—
270	C₂×C₄	—	10	1	no	—
271	C₂×C₄	D₅⋊(C₃⋊C₈)	1	1	yes	characteristic
272	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
273	C₈	—	2	1	no	—
274	C₈	—	2	1	no	—
275	C₈	—	5	1	no	—
276	C₈	—	5	1	no	—
277	C₈	—	10	1	no	—
278	C₂×C₃	—	4	1	no	—
279	C₂×C₃	—	20	1	no	—
280	C₂×C₃	#65692	1	1	yes	characteristic
281	C₂×C₃	D₅⋊M₃₂	1	1	yes	characteristic
282	C₂×C₃	#65725	1	1	yes	characteristic
283	C₅	#70858	1	1	yes	characteristic, complemented, Sylow
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	10	1	no	—
288	C₂²	—	10	1	no	—
289	C₂²	—	10	1	no	—
290	C₂²	C₂×D₅⋊(C₃⋊C₈)	1	1	yes	characteristic
291	C₄	—	2	1	no	—
292	C₄	—	5	2	no	—
293	C₄	—	10	1	no	—
294	C₄	—	10	1	no	—
295	C₄	#91990	1	1	yes	characteristic
296	C₄	D₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
297	C₃	#10603906	1	1	yes	characteristic, complemented, Sylow
298	C₂	—	4	1	no	—
299	C₂	—	20	1	no	—
300	C₂	#11752351	1	1	yes	characteristic
301	C₂	D₅⋊#812	1	1	yes	characteristic
302	C₂	#11752384	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)
commutator, socle	C₂²×C₃×C₅	C₂²×C₈
Fitting	C₃×C₅×C₂²⋊C₈	C₄
prime radical	C₂×C₃×C₂²⋊D₅	C₈
cosocle	C₂×C₃×#168	C₂³
hypercenter	C₂²⋊C₈	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#2169
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#91227	C₂	semidirect
2. C₂×#91219	C₂	semidirect
4. #11746782	C₂	semidirect
5. #11746759	C₂	semidirect
6. #11746758	C₂	semidirect
7. #11746756	C₂	semidirect
48. C₃⋊#674	D₄	semidirect
138. C₃×Dic₅	M₃₂	semidirect
222. Dic₅	#812	semidirect
242. C₃×C₅	#2169	semidirect
283. C₅	#70858	semidirect
297. C₃	#10603906	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#91227	2
C₂×#91219	2
C₃×#65014	2
#11746782	2
#11746759	2
#11746758	2
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